Formal Concept Analysis with Three Types of Negation

• URL: http://arxiv.org/abs/2608.29311v1
• Date: Sat, 29 Aug 2026 14:54:04 GMT
• Title: Formal Concept Analysis with Three Types of Negation
• Abstract summary: We introduce three types of negation concepts (contradictory negation, opposite negation, intermediary negation) into FCA.<n>Based on the set SCOI and logic LCOI+PLCOI, we define formal context, Galois connection operators, formal concept and concept lattice with three types of negation.<n>For the reasoning in FCACOI, this paper focuses on attribute implication reasoning. Based on the logic LCOI+PLCOI and its semantics, we introduce the notion of ICOI-entailment.
• Score: 0.6345523830122167
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Classic Formal Concept Analysis (FCA) primarily focuses on the positive relationships between objects and attributes and does not have mechanisms for handling negation.To overcome this limitation, we introduce three types of negation concepts (contradictory negation, opposite negation, intermediary negation) into FCA.Based on the set SCOI and logic LCOI+PLCOI with these three types negation, we define formal context, Galois connection operators, formal concept and concept lattice with three types of negation,this leads to the proposal of a FCACOI: Formal Concept Analysis with contradictory negation, opposite negation and intermediary negation.For the reasoning in FCACOI, this paper focuses on attribute implication reasoning. Based on the logic LCOI+PLCOI and its semantics, we introduce the notion of ICOI-entailment as the semantic implication for attribute implication reasoning in FCACOI. Through ICOI-entailment, a connection is established between attribute implication reasoning in FCACOI and inference in the logic LCOI+PLCOI, it indicate that formally proven inference rules (theorems) in LCOI+PLCOI are valid in the attribute implication reasoning of FCACOI, LCOI+PLCOI provides a logical foundation for attribute implication reasoning in FCACOI. To illustrate the capability of attribute implication reasoning in FCACOI, we discuss its application in a concrete example. Moreover, we explore attribute reduction of the formal context in FCACOI, propose two research frameworks for attribute reduction from different perspectives, and compare their characteristics.We believe that, based on richer logic and semantics, FCACOI elevates FCA from a theory that describes affirmations to one that can describe affirmations and its contradiction(either this or that), opposition(extreme negation) and intermediary (transitional states between oppositions).

Related papers

• Three Types of Negation of Triple and its Elements and an Extension of Triple [0.6345523830122167]
  The TCOI triple is a semantic and structural extension of the classical triple.<n>This paper explores the expressive power and reasoning of the TCOI triple, as well as the application of TCOI triple implication reasoning in counterfactuals and counterfactual reasoning.
  arXiv  Detail & Related papers  (2026-09-08T05:26:43Z)
• Concept Component Analysis: A Principled Approach for Concept Extraction in LLMs [51.378834857406325]
  Mechanistic interpretability seeks to mitigate the issues through extracts from large language models.<n>Sparse autoencoders (SAEs) have emerged as a popular approach for extracting interpretable and monosemantic concepts.<n>We show that SAEs suffer from a fundamental theoretical ambiguity: the well-defined correspondence between LLM representations and human-interpretable concepts remains unclear.
  arXiv  Detail & Related papers  (2026-01-28T09:27:05Z)
• Theory Trace Card: Theory-Driven Socio-Cognitive Evaluation of LLMs [2.98033672654447]
  We argue that many socio-cognitive evaluations proceed without an explicit theoretical specification of the target capability.<n>Without this theoretical grounding, benchmarks that exercise only narrow subsets of a capability are routinely misinterpreted as evidence of broad competence.<n>We introduce the Trace Card, a lightweight documentation artifact designed to accompany socio-cognitive evaluations.
  arXiv  Detail & Related papers  (2026-01-05T08:06:50Z)
• Extended Triangular Method: A Generalized Algorithm for Contradiction Separation Based Automated Deduction [11.18154335347018]
  This work presents the Extended Triangular Method (ETM), a generalized contradiction-construction algorithm that formalizes and extends the internal mechanisms of contradiction separation.<n>By bridging theoretical abstraction and operational implementation, ETM advances the contradiction separation paradigm into a generalized, scalable, and practically competitive model for automated reasoning.
  arXiv  Detail & Related papers  (2025-10-12T17:03:33Z)
• Explainable Chain-of-Thought Reasoning: An Empirical Analysis on State-Aware Reasoning Dynamics [69.00587226225232]
  We introduce a state-aware transition framework that abstracts CoT trajectories into structured latent dynamics.<n>To characterize the global structure of reasoning, we model their progression as a Markov chain.<n>This abstraction supports a range of analyses, including semantic role identification, temporal pattern visualization, and consistency evaluation.
  arXiv  Detail & Related papers  (2025-08-29T18:53:31Z)
• Three Kinds of Negation in Knowledge and Their Mathematical Foundations [4.343358400426713]
  We examine and analyze the understanding and characteristics of negation in various fields such as philosophy, logic, and linguistics.<n>Based on the distinction between the concepts of contradiction and opposition, we propose that there are three different types of negation in knowledge from a conceptual perspective.
  arXiv  Detail & Related papers  (2025-05-30T10:01:37Z)
• Rational Inference in Formal Concept Analysis [0.4499833362998487]
  Defeasible conditionals are a form of non-monotonic inference.<n>KLM framework defines a semantics for the propositional case of defeasible conditionals.
  arXiv  Detail & Related papers  (2025-04-07T20:15:20Z)
• The Curse of CoT: On the Limitations of Chain-of-Thought in In-Context Learning [56.574829311863446]
  Chain-of-Thought (CoT) prompting has been widely recognized for its ability to enhance reasoning capabilities in large language models (LLMs)<n>We demonstrate that CoT and its reasoning variants consistently underperform direct answering across varying model scales and benchmark complexities.<n>Our analysis uncovers a fundamental hybrid mechanism of explicit-implicit reasoning driving CoT's performance in pattern-based ICL.
  arXiv  Detail & Related papers  (2025-04-07T13:51:06Z)
• Modeling Hierarchical Reasoning Chains by Linking Discourse Units and Key Phrases for Reading Comprehension [80.99865844249106]
  We propose a holistic graph network (HGN) which deals with context at both discourse level and word level, as the basis for logical reasoning. Specifically, node-level and type-level relations, which can be interpreted as bridges in the reasoning process, are modeled by a hierarchical interaction mechanism.
  arXiv  Detail & Related papers  (2023-06-21T07:34:27Z)
• Logical Satisfiability of Counterfactuals for Faithful Explanations in NLI [60.142926537264714]
  We introduce the methodology of Faithfulness-through-Counterfactuals. It generates a counterfactual hypothesis based on the logical predicates expressed in the explanation. It then evaluates if the model's prediction on the counterfactual is consistent with that expressed logic.
  arXiv  Detail & Related papers  (2022-05-25T03:40:59Z)
• Rediscovering Argumentation Principles Utilizing Collective Attacks [26.186171927678874]
  We extend the principle-based approach to Argumentation Frameworks with Collective Attacks (SETAFs) Our analysis shows that investigating principles based on decomposing the given SETAF (e.g. directionality or SCC-recursiveness) poses additional challenges in comparison to usual AFs.
  arXiv  Detail & Related papers  (2022-05-06T11:41:23Z)
• Causal Inference Principles for Reasoning about Commonsense Causality [93.19149325083968]
  Commonsense causality reasoning aims at identifying plausible causes and effects in natural language descriptions that are deemed reasonable by an average person. Existing work usually relies on deep language models wholeheartedly, and is potentially susceptible to confounding co-occurrences. Motivated by classical causal principles, we articulate the central question of CCR and draw parallels between human subjects in observational studies and natural languages. We propose a novel framework, ROCK, to Reason O(A)bout Commonsense K(C)ausality, which utilizes temporal signals as incidental supervision.
  arXiv  Detail & Related papers  (2022-01-31T06:12:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.